articles Calculating the surgically induced refractive change following ocular surgery

Jack T. Holladay, M.D., Thomas V. Cravy, M.D., Douglas D. Koch, M.D.

Calculating the surgically induced spherical and astigmatic change is required to evaluate existing and evolving corneal surgical techniques. A variety of formulas to calculate corneal astigmatic changes have been developed. All represent gross simpli­fi<.!ations of the complex topographical changes that occur on the aspheric corneal surface. Neverthe­less, these formulas provide extremely useful data that assist in planning and evaluating corneal and limbal surgery.

The classic formula for calculating the surgically induced refractive change (SIRe) was described over 100 years ago. 1 It assumes that the induced corneal spherical and astigmatic change can be rep­resented by a sphere and cylinder that, when placed in front of the eye at the corneal vertex,

would produce the same optical effect as the surgery.2 Some 125 years later, Jaffe and Clayman described three trigonometric methods for per­forming these calculations. 3 The fundamental ad­vantage of this approach is its inherent consistency between refractive and keratometric changes and its sound mathematical basis.

Cravy adopted a different approach and devel­oped a formula based solely upon keratometric values that elegantly facilitated the aggregate analysis of groups of patients. 4 Cravy's formula has been widely used to calculate post-cataract sur­gery astigmatic changes, as it permits the calcula­tion of aggregate with-the-rule or against-the-rule changes. To address some of the inconsistencies in Cravy's formula, Koch and Russell developed a for-

From Hermann Eye Center, University of Texas Medical School at Houston (Holladay), Cullen Eye Institute, Baylor College of Medicine, Houston (Koch), and Santa Maria, California (Cravy).

Reprint requests to Jack T. Holladay, M.D., Hermann Eye Center, 6411 Fannin, Houston, Texas 77030.

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mula that was derived from his method (T. Russell, "A New Formula for Calculating Changes in Cor­neal Astigmatism," Symposium on Cataract, IOL, and Refractive Surgery, Boston, April 1991).

Although each of these formulas has unique mer­its, we believe it is important to establish a stan­dardized method for calculating and reporting spherical and astigmatic changes. We therefore propose that a straightforward, ten-step method based on the oblique cross-cylinder solution be adopted as this standard, because it is based on measurable optical changes and integrates refrac­tive and keratometric changes. We will present the mathematical basis of this method and its clinical applications, and give several examples that calcu­late the entire SIRC following ocular surgery. Adopting these techniques will result in uniform, consistent, and understandable reporting of results by different investigators.

METHODS

Mathematical Solution to Obliquely Crossed Spherocylinders

There are four published methods for obliquely crossed spherocylinders: rectangular coordinate method, graphic vector method, matrix method, and the law of sines and cosines method for the cylinder change.3 ,5 Each of these methods accom­plishes the same task in a different way, using trig­onometric identities which yield the same unique result for any single pair of obliquely crossed sphe­rocylinders. We will present a modification of the rectangular coordinate method in ten steps, which is simple to perform by hand or to implement on a programmable calculator or computer.

Determine the resultant Spherocylinder 3 from obliquely crossed cylinders Spherocylinder 1 and Spherocylinder 2.

Given: Spherocylinder 1 = SCI where SI = sphere of SCI

Cl = cylinder of SCI Al = axis of cylinder Cl

Spherocylinder 2 = SC2 where S2 = sphere of SC2

C2 = cylinder of SC2 A2 = axis of cylinder C2

Find: Spherocylinder 3 = SC3 where S3 = sphere of SC3

C3 = cylinder of SC3 A3 = axis of cylinder C3

Step l. Transpose SCi and SC2 so their cylinders have the same sign.

Step 2. SCi must be chosen so the value of Ai is smaller than A2.

Step 3. Find angle a, the difference between A2 and Ai.

IX = A2 - Al

Note: a must be positive. Step 4. Find angle 2 {3 from the formula:

Tan 2 {3 = C2 sin 2 IX

Cl + C2 cos 2 IX

The denominator can sometimes be zero, which "blows up" on most computers that cannot divide by zero. The whole term actually approaches in­finity when the denominator approaches zero, which results in 2 {3 equaling 900. A simple pro­gramming solution is to add a very small value to the denominator such as 0.0000000001.

Step 5. Find angle (J from the following formula:

(J = (2 (3 + 180°) 2

Step 6. Determine the sphere contributed by the two cross cylinders (SC) from

sc = Cl sin2 (J + C2 sin2 (IX - (J)

Step 7. Determine the total spherical result (S3) from the following formula:

S3 = SI + S2 + SC

Step 8. Determine the total cylindrical result (C3) from the following formula:

C3 = Cl + C2 - 2 SC

Step 9. Determine the resultant axis (A3) in stan­dard notation from the following formula:

A3 = Al + (J

If A3 is greater than 180°, subtract 180° for stan­dard axis notation; if A3 is negative, add 180°.

Step 10. Any spherocylinder (SC3) can be writ­ten in one of three forms: the plus or minus sphero­cylinder form and the cross cylinder form. The alternate spherocylindrical form (SC4) and cross cylinder form (XC5) are calculated in the following manner using the transposition rules.

A. Alternate spherocylindrical form (SC4) of SC3: S4 = S3 + C3 = sphere of SC4 C4 = - C3 = cylinder of SC4 A4 = A3 ± 90° = axis of SC4

B. Cross cylinder form (XC5) of SC3:

C5A = S3 = cross cylinder A of XC5 A5A = A3 ± 90° = axis of cross cylinder A of XC5 C5B = S3 + C3 = cross cylinder B of XC5 A5B = A3 = axis of cross cylinder B of XC5

Although each of these three forms repre­sents the exact same spherocylinder, we will

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see that each form has specific benefits in visual­izing the effect of the surgical procedure being evaluated.

I. APPLICATION 1. Adding over-refraction to spectacles.

Example lA. Write the patient's final spectacle prescription given the spectacle Rx and the over-refraction, where

Equation 1: Final Rx = Old Rx + Over-refraction

Given: Old spectacle Rx: sphere = 10.00 D cylinder = +2.00 D

axis = 180°

Over-refraction: sphere = +2.25 D cylinder = + 1.25 D

axis = 45°

STEP 1. Both spherocylinders are already in plus cylin­der form.

STEP 2. The over-refraction has the smallest axis and is therefore chosen as SCI.

OldRx Over-refraction

SI = +2.25 Cl = +1.25 Al = 45°

S2 = +10.00 C2 = +2.00 A2 = 1800

STEP 3. a = A2 - Al = 180° - 45°

a = 135°

STEP 4.

C2 sin 2 a Tan 2 fJ = Cl + C2 cos 2 a

+2 * sin(2700) +2 * (-1) Tan 2 fJ = +1.25 + 2 * cos(2700)

-2 +1.25 + 2 * (0)

Tan 2 fJ = +1.25 = -1.60

2 fJ = -58°

STEP 5.

8 = (2 fJ + 1800) . 2

8 = (-58 + 180°) = 122° = 61 ° 2 2

STEP 6. SC = Cl sin2 8 + C2 sin2 (a - 8) = 1.25 * (sin 61°)2 + 2 * [sin(135° - 61°)]2 = 1.25 * (.875)2 + 2 * (.961)2

.956 + 1.85 SC = +2.80

STEP 7. S3 = SI + S2 + SC = +2.25 + 10.00 + 2.80

S3 = +15.05 STEP 8. C3 = Cl + C2 - 2SC

= +1.25 + 2.00 - 2 * (+2.80) C3 = -2.36

STEP 9. A3 = Al + 8 = 45° + 61 °

A3 = 106°

STEP 10. A. Alternate spherocylindrical form (SC4):

S4 = S3 + C3 = +15.05 + (-2.36) = +12.70 C4 = -C3 = - (-2.36) = +2.36 A4 = A3 ± 90° = 106° - 90° = 16°

B. Cross cylinder form (XC5): C5A= S3 = +15.05 A5A = A3 ± 90° = 106° - 90° = 16° C5B = S3 + C3 = +15.05 + (-2.36) = +12.70 A5B = A3 = 106°

The patient's final prescription written in the three stan-dard axis forms would be

plus cyl form : +12.70 + 2.36 X 16° Minus cyl form: +15.05 - 2.36 X 106° Cross cyl form: + 15.05 X 16° and + 12.70 X 106°

This same method can be used for refining toric soft contact lens prescriptions using an over­refraction. 6

II. APPLICATION 2. Determining the SIRC from the preoperative refraction (Rpre) and the postop­erative refraction (Rpost).

Before applying the obliquely crossed cylinder solution we must review some basic principles of refraction. First, the basic principle is that the error of the eye (EE) plus the optical correction (Rx) must equal the residual error (RE). Expressed algebra­ically this would be

Equation A: EE + Rx = RE

If there is no residual error, i.e., the optical cor­rection is exactly correct, then RE is 0, and the error of the eye is exactly equal to the negative of the optical correction.

Equation B: EE = - Rx, when RE = 0

For spherical refractive errors this is readily ap­parent. If a patient is myopic by 1 diopter (D), the optical prescription is -1.00 D and the error of the eye is + 1.00 D. The plus sign simply indicates that the eye has an excess power of 1.00 D. This same relationship holds true for spherocylindrical re­fractive errors. One needs only to change the signs of the optical correction to determine the error of the eye. For example,

If Rx = -1.00 + 1.00 X 90° then EE = +1.00 - 1.00 X 90°.

Only the signs change; the axis remains the same. Now, using equation A,

EE+Rx=RE

we can let EE be the preoperative error of the eye and RE be the residual error of the eye after sur­gery. Rx is the "correction" or the SIRC that has

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